Bridging the Vector Calculus Gap

数学科学物理学在数学家通常教授向量微积分的方法和其他科学家使用向量微积分的方法之间存在着“向量微积分差距”。这种材料对物理学家和一些工程师来说是必不可少的，因为它在描述电和磁方面起着核心作用。但是，数学家用来教授这些材料的传统语言与应用程序中使用的方式如此不同，以至于学生们常常无法翻译。问题的一个主要部分是传统数学强调笛卡尔坐标将向量描述为数字的三元组，而不是强调向量是空间中的箭头。这导致点积和叉积被记忆为代数公式，而不是分别关于投影和面积的陈述。传统方法有一个很大的优点，那就是为处理相当普遍的问题提供一个单一的框架。但大多数实际应用，包括几乎所有本科阶段的应用，都属于少数特殊情况，例如球面或圆柱对称。当计算在适当的坐标下完成时，高度对称的问题变得更加直观，使用适应这些坐标的向量基。这强调了问题的几何性质，而不是蛮力的代数计算。这个项目正在开发补充材料，特别是小组活动，这些材料强调高度对称情况下的几何，其中一些用于传统的向量微积分课程，另一些用于新的高年级物理课程中相关材料。这些活动向学生介绍他们在选择的专业中遇到的问题类型和解决方法，同时增加他们对传统向量微积分及其应用的理解，从而弥合向量微积分的差距。

Mathematical Sciences (21) Physics (13)There is a "vector calculus gap" between the way vector calculus is usually taught by mathematicians and the way it is used by other scientists. This material is essential for physicists and some engineers due to its central role in the description of electricity and magnetism. But the traditional language used by mathematicians to teach this material is so different from the way it is used in applications that students are often unable to translate. A major part of the problem is the traditional mathematics emphasis on Cartesian coordinates to describe vectors as triples of numbers, rather than emphasizing that vectors are arrows in space. This leads to the dot and cross products being memorized as algebraic formulas, rather than statements about projections and areas, respectively. The traditional approach has the one big advantage of providing a single framework for handling quite general problems. But most practical applications, including virtually all at the undergraduate level, fall into a small number of special cases, such as those with spherical or cylindrical symmetry. Problems with a high degree of symmetry become much more intuitive when the computations are done in appropriate coordinates, using a vector basis adapted to those coordinates. This emphasizes the geometry of the problem, rather than a brute force algebraic computation.This project is developing supplemental materials, especially small group activities, which emphasize the geometry of highly symmetric situations, some of which are intended for use with an otherwise traditional vector calculus course, and some of which are intended for use in a new, upper-division physics course on related material. Such activities introduce students to the types of problems and methods of solution which they encounter in their chosen specialization, while at the same time increasing their understanding of traditional vector calculus and its applications, thus bridging the vector calculus gap.